#501391

order	512
nilpotency class	4
isoclinism	isoclinic class of #501052
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#32942	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

683 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,17,27,16,19,28,33,14,12,15,20,13,21,30,29)(11,22,32,35,31,23,37,39,25,18,26,36,24,34,40,38);(2,5)(6,8)(9,11)(10,18)(12,22)(13,24)(14,25)(15,26)(16,31)(17,32)(19,34)(20,35)(21,23)(27,36)(28,37)(29,39)(30,40)(33,38);(1,3,4,7)(2,6,5,8)(9,12,14,10)(11,23,25,34)(13,21,16,19)(15,29,17,33)(18,31,22,24)(20,28,27,30)(26,36,32,35)(37,38,40,39);(2,5)(6,8)(9,13,14,16)(10,19,12,21)(11,24,25,31)(15,28,17,30)(18,34,22,23)(20,29,27,33)(26,37,32,40)(35,39,36,38)

Polycyclic presentation:

2^9|a^2=fi,c^2=e,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408587);